Generalization of bi-canonical degrees

TL;DR

This paper extends the concept of bi-canonical degrees to a broader class of Cohen-Macaulay rings, including those without canonical modules, providing new invariants to measure their deviation from Gorenstein rings.

Contribution

It generalizes the bi-canonical degree to rings where the canonical module is not an ideal and explores related invariants for rings lacking canonical modules.

Findings

Extended bi-canonical degree to more general rings.

Provided new invariants for rings without canonical modules.

Enhanced understanding of Cohen-Macaulay ring deviations from Gorenstein property.

Abstract

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $\omega$. Attached to each such ring R, when $\omega$ is an ideal, there are integers--the type of R, the reduction number of $\omega$--that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In arXiv:1701.05592 and arXiv:1711.09480 we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where $\omega$ is not necessarily an ideal. We also discuss generalizations to rings without canonical modules but admitting modules sharing some of their properties.